[Paper Review] Randomised Wasserstein Barycenter Computation: Resampling with Statistical Guarantees

Florian Heinemann, Axel Munk+1 more|arXiv (Cornell University)|Dec 11, 2020

Topological and Geometric Data Analysis63 references4 citations

TL;DR

This paper proposes a randomized resampling method to compute Wasserstein barycenters on large-scale datasets with nonasymptotic error bounds that balance computational cost and statistical accuracy. The method combines any exact solver with resampling to achieve optimal error rates independent of dimension, demonstrated on simulated and real microscopy data beyond current state-of-the-art reach.

ABSTRACT

We propose a hybrid resampling method to approximate finitely supported Wasserstein barycenters on large-scale datasets, which can be combined with any exact solver. Nonasymptotic bounds on the expected error of the objective value as well as the barycenters themselves allow to calibrate computational cost and statistical accuracy. The rate of these upper bounds is shown to be optimal and independent of the underlying dimension, which appears only in the constants. Using a simple modification of the subgradient descent algorithm of Cuturi and Doucet, we showcase the applicability of our method on a myriad of simulated datasets, as well as a real-data example from cell microscopy which are out of reach for state of the art algorithms for computing Wasserstein barycenters.

Motivation & Objective

• To address the computational intractability of Wasserstein barycenter computation on large datasets by introducing a resampling framework with statistical guarantees.
• To provide nonasymptotic error bounds on both the objective value and the barycenter itself, enabling calibration of computational effort and statistical precision.
• To demonstrate applicability on high-dimensional, complex data such as cell microscopy images, where existing algorithms fail.
• To show that the error rate is optimal and independent of ambient dimension, with dimension only affecting constants.

Proposed method

• The method introduces a hybrid resampling scheme that approximates finitely supported Wasserstein barycenters by subsampling data measures while preserving statistical accuracy.
• It combines any exact barycenter solver with a randomized resampling strategy to reduce computational cost while maintaining theoretical error bounds.
• Nonasymptotic upper bounds on the expected error of the objective value and the barycenter are derived, showing optimal convergence rates independent of dimension.
• The approach leverages a modified subgradient descent algorithm (from Cuturi and Doucet) to efficiently compute barycenters on large-scale datasets.
• Theoretical analysis establishes that the error bound depends on a constant $ C_P $, derived from the minimum ratio of cost difference to distance from the optimal set, ensuring positivity and convergence.
• The method relates the $ L^1 $-distance between transport plans to the total variation and Wasserstein distance, enabling error quantification via $ W_p^p(\mu, \mu^*) $.

Experimental results

Research questions

• RQ1 Can a randomized resampling method be designed to approximate Wasserstein barycenters on large-scale datasets with theoretical error guarantees?
• RQ2 What is the optimal rate of convergence for the expected error in the objective value and barycenter, and does it depend on the ambient dimension?
• RQ3 Can the method be combined with existing exact solvers to maintain accuracy while reducing computational cost?
• RQ4 How does the method perform on real-world high-dimensional data, such as cell microscopy images, compared to state-of-the-art algorithms?
• RQ5 Is the derived error bound tight, and does it achieve the optimal rate under minimal assumptions?

Key findings

• The proposed method achieves nonasymptotic error bounds on the expected objective value and barycenter that are optimal and independent of the underlying dimension, with dimension only affecting constants.
• The error rate is shown to be optimal, with the bound depending on a positive constant $ C_P $ derived from the minimum ratio of cost difference to distance from the optimal set of transport plans.
• The method enables computation of Wasserstein barycenters on datasets that are computationally infeasible for current state-of-the-art algorithms, including real microscopy data.
• A modified subgradient descent algorithm is successfully applied to large-scale simulated and real datasets, demonstrating practical scalability.
• Theoretical analysis confirms that the $ L^1 $-distance between transport plans lower-bounds the Wasserstein distance, enabling tight error control via total variation.
• The positivity of the constant $ C_P $ is rigorously established, ensuring the error bound is non-vacuous and meaningful for convergence analysis.